Composition theorems in communication complexity

In the Boolean block composed function case, the regularity condition reduces to the
matrix [g(x, y)] being balanced, and later we will prove that the orthogonality condition
reduces to the strongly balanced property. From this theorem we can see that the way to
partition Ψ into Ψ Easy and Ψ Hard does not really matter for the lower bound proof passing
through. However, the partition does play a role when we later bound the spectral norm in
the denominator.
5.2 Functions with group symmetry
For a general finite group G, two elements s and t are conjugate, denoted by s ∼ t, if there
exists an element r ∈ G s.t. rsr −1 = t. Define H as the set of all class functions, i.e. functions
f s.t. f(s) = f(t) if s ∼ t. Then H is an h-dimensional subspace of L C (G), where h is the
number of conjugacy classes. The irreducible characters {χ i : i ∈ [h]} form an orthogonal
basis of H. For a class function f and irreducible characters χ i , denote by ˆf i the coefficient
of χ i in expansion of f according to χ i ’s, i.e. ˆfi = 〈χ i , f〉 = 1
|G|
fact is that for any i, we have
| ˆf i | = 1
|G|
∑
χ i (g)f(g)
∣
∣ ≤ 1 ∑
|f(g)||χ i (g)| ≤
|G|
g∈G
g∈G
( 1
|G|
∑
g∈G
∑
g∈G χ i(g)f(g). An easy
)
|f(g)|
· max |χ i (g)|. (5)
g
If G is Abelian, then it always has |χ i (g)| = 1, thus max i | ˆf i | ≤ 1 ∑
|G| g∈G
|f(g)|. For general
groups, we have |χ i (g)| ≤ deg(χ i ), where deg(χ i ) is the degree of χ i , namely the dimension
of the associated vector space.
In this section we consider the setting that S is a finite group G. The goal is to exploit
properties of group characters to give better form of the lower bound. In particular, we hope
to see when the second condition holds and what the matrix operator norm ‖[ψ ( g(x, y))] x,y ‖
is in this setting.
The standard orthogonality of irreducible characters says that ∑ s∈G χ i(s)χ j (s) = 0. The
second condition in Theorem 11 is concerned with a more general case: For a multiset T with
elements in G × G, we need
∑
χ i (s)χ j (t) = 0, ∀i ≠ j. (6)
(s,t)∈T
The standard orthogonality relation corresponds to the special that T = {(s, s) : s ∈ G}. We
hope to have a characterization of a multiset T to make Eq. (6) hold.
We may think of the a multiset T with elements in set S as a function on S, with the
value on s ∈ S being the multiplicity of s in T . Since characters are class functions, for each
pair (C k , C l ) of conjugacy classes, only the value ∑ g 1∈C k ,t∈C l
T (g 1 , t) matters for the sake of
Eq. (6). We thus make T a class function by taking average within each class pair (C k , C l ).
That is, define a new function T ′ as
T ′ (s, t) =
∑
T (s, t)/(|C k ||C l |), ∀s ∈ C k , ∀t ∈ C l .
s∈C k ,t∈C l
Proposition 2. For a finite group G and a multiset T with elements in G×G, the following
three statements are equivalent:
1. ∑ (s,t)∈T χ i(s)χ j (t) = 0, ∀i ≠ j
2. T ′ , as a function, is in span{χ i ⊗ χ i : i ∈ [h]}
3. [T ′ (s, t)] s,t = C † DC where D is a diagonal matrix and C = [χ i (s)] i,s . That is, T ′ , as a
matrix, is normal and diagonalized exactly by the irreducible characters.